Chapter 12: Compressible Flow

1
Chapter 12 Compressible Flow

• Eric G. Paterson
• Department of Mechanical and Nuclear Engineering
• The Pennsylvania State University
• Spring 2005

2
Note to Instructors

• These slides were developed1, during the spring
  semester 2005, as a teaching aid for the
  undergraduate Fluid Mechanics course (ME33
  Fluid Flow) in the Department of Mechanical and
  Nuclear Engineering at Penn State University.
  This course had two sections, one taught by
  myself and one taught by Prof. John Cimbala.
  While we gave common homework and exams, we
  independently developed lecture notes. This was
  also the first semester that Fluid Mechanics
  Fundamentals and Applications was used at PSU.
  My section had 93 students and was held in a
  classroom with a computer, projector, and
  blackboard. While slides have been developed
  for each chapter of Fluid Mechanics
  Fundamentals and Applications, I used a
  combination of blackboard and electronic
  presentation. In the student evaluations of my
  course, there were both positive and negative
  comments on the use of electronic presentation.
  Therefore, these slides should only be integrated
  into your lectures with careful consideration of
  your teaching style and course objectives.
• Eric Paterson
• Penn State, University Park
• August 2005

1 This Chapter was not covered in our class.
These slides have been developed at the request
of McGraw-Hill
3
Objectives

• Appreciate the consequences of compressibility in
  gas flows
• Understand why a nozzle must have a diverging
  section to accelerate a gas to supersonic speeds
• Predict the occurrence of shocks and calculate
  property changes across a shock wave
• Understand the effects of friction and heat
  transfer on compressible flows

4
Stagnation Properties

• Recall definition of enthalpy
• which is the sum of internal energy u and flow
  energy P/?
• For high-speed flows, enthalpy and kinetic energy
  are combined into stagnation enthalpy h0

5
Stagnation Properties

• Steady adiabatic flow through duct with no
  shaft/electrical work and no change in elevation
  and potential energy
• Therefore, stagnation enthalpy remains constant
  during steady-flow process

6
Stagnation Properties

• If a fluid were brought to a complete stop (V2
  0)
• Therefore, h0 represents the enthalpy of a fluid
  when it is brought to rest adiabatically.
• During a stagnation process, kinetic energy is
  converted to enthalpy.
• Properties at this point are called stagnation
  properties (which are identified by subscript 0)

7
Stagnation Properties

• If the process is also reversible, the stagnation
  state is called the isentropic stagnation state.
• Stagnation enthalpy is the same for isentropic
  and actual stagnation states
• Actual stagnation pressure P0,act is lower than
  P0 due to increase in entropy s as a result of
  fluid friction.
• Nonetheless, stagnation processes are often
  approximated to be isentropic, and isentropic
  properties are referred to as stagnation
  properties

8
Stagnation Properties

• For an ideal gas, h CpT, which allows the h0 to
  be rewritten
• T0 is the stagnation temperature. It represents
  the temperature an ideal gas attains when it is
  brought to rest adiabatically.
• V2/2Cp corresponds to the temperature rise, and
  is called the dynamic temperature
• For ideal gas with constant specific heats,
  stagnation pressure and density can be expressed
  as

9
Stagnation Properties

• When using stagnation enthalpies, there is no
  need to explicitly use kinetic energy in the
  energy balance.
• Where h01 and h02 are stagnation enthalpies at
  states 1 and 2.
• If the fluid is an ideal gas with constant
  specific heats

10
Speed of Sound and Mach Number

• Important parameter in compressible flow is the
  speed of sound.
• Speed at which infinitesimally small pressure
  wave travels
• Consider a duct with a moving piston
• Creates a sonic wave moving to the right
• Fluid to left of wave front experiences
  incremental change in properties
• Fluid to right of wave front maintains original
  properties

11
Speed of Sound and Mach Number

• Construct CV that encloses wave front and moves
  with it
• Mass balance

Neglect H.O.T.
cancel
12
Speed of Sound and Mach Number

• Energy balance ein eout

Neglect H.O.T.
cancel
cancel
13
Speed of Sound and Mach Number

• Using the thermodynamic relation
• Combing this with mass and energy conservation
  gives
• For an ideal gas

14
Speed of Sound and Mach Number

• Since
• R is constant
• k is only a function of T
• Speed of sound is only a function of temperature

15
Speed of Sound and Mach Number

• Second important parameter is the Mach number Ma
• Ratio of fluid velocity to the speed of sound
• Flow regimes classified in terms of Ma

Ma lt 1 Subsonic Ma 1 Sonic Ma gt 1
Supersonic Ma gtgt 1 Hypersonic Ma ? 1 Transonic
16
One-Dimensional Isentropic Flow

• For flow through nozzles, diffusers, and turbine
  blade passages, flow quantities vary primarily in
  the flow direction
• Can be approximated as 1D isentropic flow
• Consider example of Converging-Diverging Duct

17
One-Dimensional Isentropic Flow

• Example 12-3 illustrates
• Ma 1 at the location of the smallest flow area,
  called the throat
• Velocity continues to increase past the throat,
  and is due to decrease in density
• Area decreases, and then increases. Known as a
  converging - diverging nozzle. Used to
  accelerate gases to supersonic speeds.

18
One-Dimensional Isentropic Flow Variation of
Fluid Velocity with Flow Area

• Relationship between V, ?, and A are complex
• Derive relationship using continuity, energy,
  speed of sound equations
• Continuity
• Differentiate and divide by mass flow rate (?AV)

19
One-Dimensional Isentropic Flow Variation of
Fluid Velocity with Flow Area

• Derived relation (on image at left) is the
  differential form of Bernoullis equation.
• Combining this with result from continuity gives
• Using thermodynamic relations and rearranging

20
One-Dimensional Isentropic Flow Variation of
Fluid Velocity with Flow Area

• This is an important relationship
• For Ma lt 1, (1 - Ma2) is positive ? dA and dP
  have the same sign.
• Pressure of fluid must increase as the flow area
  of the duct increases, and must decrease as the
  flow area decreases
• For Ma gt 1, (1 - Ma2) is negative ? dA and dP
  have opposite signs.
• Pressure must increase as the flow area
  decreases, and must decrease as the area increases

21
One-Dimensional Isentropic Flow Variation of
Fluid Velocity with Flow Area

• A relationship between dA and dV can be derived
  by substituting ?V -dP/dV (from the
  differential Bernoulli equation)
• Since A and V are positive
• For subsonic flow (Ma lt 1) dA/dV lt 0
• For supersonic flow (Ma gt 1) dA/dV gt 0
• For sonic flow (Ma 1) dA/dV 0

22
One-Dimensional Isentropic Flow Variation of
Fluid Velocity with Flow Area
Comparison of flow properties in subsonic and
supersonic nozzles and diffusers
23
One-Dimensional Isentropic Flow Property
Relations for Isentropic Flow of Ideal Gases

• Relations between static properties and
  stagnation properties in terms of Ma are useful.
• Earlier, it was shown that stagnation temperature
  for an ideal gas was
• Using definitions, the dynamic temperature term
  can be expressed in terms of Ma

24
One-Dimensional Isentropic Flow Property
Relations for Isentropic Flow of Ideal Gases

• Substituting T0/T ratio into P0/P and ?0/?
  relations (slide 8)
• Numerical values of T0/T, P0/P and ?0/? compiled
  in Table A-13 for k1.4
• For Ma 1, these ratios are called critical
  ratios

25
One-Dimensional Isentropic Flow Property
Relations for Isentropic Flow of Ideal Gases
26
Isentropic Flow Through Nozzles

• Converging or converging-diverging nozzles are
  found in many engineering applications
• Steam and gas turbines, aircraft and spacecraft
  propulsion, industrial blast nozzles, torch
  nozzles
• Here, we will study the effects of back pressure
  (pressure at discharge) on the exit velocity,
  mass flow rate, and pressure distribution along
  the nozzle

27
Isentropic Flow Through NozzlesConverging Nozzles

• State 1 Pb P0, there is no flow, and pressure
  is constant.
• State 2 Pb lt P0, pressure along nozzle
  decreases.
• State 3 Pb P , flow at exit is sonic, creating
  maximum flow rate called choked flow.
• State 4 Pb lt Pb, there is no change in flow or
  pressure distribution in comparison to state 3
• State 5 Pb 0, same as state 4.

28
Isentropic Flow Through NozzlesConverging Nozzles

• Under steady flow conditions, mass flow rate is
  constant
• Substituting T and P from the expressions on
  slides 23 and 24 gives
• Mass flow rate is a function of stagnation
  properties, flow area, and Ma

29
Isentropic Flow Through NozzlesConverging Nozzles

• The maximum mass flow rate through a nozzle with
  a given throat area A is fixed by the P0 and T0
  and occurs at Ma 1
• This principal is important for chemical
  processes, medical devices, flow meters, and
  anywhere the mass flux of a gas must be known and
  controlled.

30
Isentropic Flow Through NozzlesConverging-Divergi
ng Nozzles

• The highest velocity in a converging nozzle is
  limited to the sonic velocity (Ma 1), which
  occurs at the exit plane (throat) of the nozzle
• Accelerating a fluid to supersonic velocities (Ma
  gt 1) requires a diverging flow section
• Converging-diverging (C-D) nozzle
• Standard equipment in supersonic aircraft and
  rocket propulsion
• Forcing fluid through a C-D nozzle does not
  guarantee supersonic velocity
• Requires proper back pressure Pb

31
Isentropic Flow Through NozzlesConverging-Divergi
ng Nozzles

• P0 gt Pb gt Pc
• Flow remains subsonic, and mass flow is less than
  for choked flow. Diverging section acts as
  diffuser
• Pb PC
• Sonic flow achieved at throat. Diverging section
  acts as diffuser. Subsonic flow at exit.
  Further decrease in Pb has no effect on flow in
  converging portion of nozzle

32
Isentropic Flow Through NozzlesConverging-Divergi
ng Nozzles

• PC gt Pb gt PE
• Fluid is accelerated to supersonic velocities in
  the diverging section as the pressure decreases.
  However, acceleration stops at location of normal
  shock. Fluid decelerates and is subsonic at
  outlet. As Pb is decreased, shock approaches
  nozzle exit.
• PE gt Pb gt 0
• Flow in diverging section is supersonic with no
  shock forming in the nozzle. Without shock, flow
  in nozzle can be treated as isentropic.

33
Shock Waves and Expansion Waves

• Review
• Sound waves are created by small pressure
  disturbances and travel at the speed of sound
• For some back pressures, abrupt changes in fluid
  properties occur in C-D nozzles, creating a shock
  wave
• Here, we will study the conditions under which
  shock waves develop and how they affect the flow.

34
Shock Waves and Expansion WavesNormal Shocks

• Shocks which occur in a plane normal to the
  direction of flow are called normal shock waves
• Flow process through the shock wave is highly
  irreversible and cannot be approximated as being
  isentropic
• Develop relationships for flow properties before
  and after the shock using conservation of mass,
  momentum, and energy

35
Shock Waves and Expansion WavesNormal Shocks

• Conservation of mass
• Conservation of energy
• Conservation of momentum
• Increase in entropy

36
Shock Waves and Expansion WavesNormal Shocks

• Combine conservation of mass and energy into a
  single equation and plot on h-s diagram
• Fanno Line locus of states that have the same
  value of h0 and mass flux
• Combine conservation of mass and momentum into a
  single equation and plot on h-s diagram
• Rayleigh line
• Points of maximum entropy correspond to Ma 1.
• Above / below this point is subsonic / supersonic

37
Shock Waves and Expansion WavesNormal Shocks

• There are 2 points where the Fanno and Rayleigh
  lines intersect points where all 3 conservation
  equations are satisfied
• Point 1 before the shock (supersonic)
• Point 2 after the shock (subsonic)
• The larger Ma is before the shock, the stronger
  the shock will be.
• Entropy increases from point 1 to point 2
  expected since flow through the shock is
  adiabatic but irreversible

38
Shock Waves and Expansion WavesNormal Shocks

• Equation for the Fanno line for an ideal gas with
  constant specific heats can be derived
• Similar relation for Rayleigh line is
• Combining this gives the intersection points

39
Shock Waves and Expansion WavesOblique Shocks

• Not all shocks are normal to flow direction.
• Some are inclined to the flow direction, and are
  called oblique shocks

40
Shock Waves and Expansion WavesOblique Shocks

• At leading edge, flow is deflected through an
  angle ? called the turning angle
• Result is a straight oblique shock wave aligned
  at shock angle ? relative to the flow direction
• Due to the displacement thickness, ??is slightly
  greater than the wedge half-angle ?.

41
Shock Waves and Expansion WavesOblique Shocks

• Like normal shocks, Ma decreases across the
  oblique shock, and are only possible if upstream
  flow is supersonic
• However, unlike normal shocks in which the
  downstream Ma is always subsonic, Ma2 of an
  oblique shock can be subsonic, sonic, or
  supersonic depending upon Ma1 and ?.

42
Shock Waves and Expansion WavesOblique Shocks

• All equations and shock tables for normal shocks
  apply to oblique shocks as well, provided that we
  use only the normal components of the Mach number
• Ma1,n V1,n/c1
• Ma2,n V2,n/c2

???-Ma relationship
43
Shock Waves and Expansion WavesOblique Shocks
44
Shock Waves and Expansion WavesOblique Shocks

• If wedge half angle ? gt ?max, a detached oblique
  shock or bow wave is formed
• Much more complicated that straight oblique
  shocks.
• Requires CFD for analysis.

45
Shock Waves and Expansion WavesOblique Shocks

• Similar shock waves see for axisymmetric bodies,
  however, ???-Ma relationship and resulting
  diagram is different than for 2D bodies

46
Shock Waves and Expansion WavesOblique Shocks

• For blunt bodies, without a sharply pointed nose,
  ? 90?, and an attached oblique shock cannot
  exist regardless of Ma.

47
Shock Waves and Expansion WavesPrandtl-Meyer
Expansion Waves

• In some cases, flow is turned in the opposite
  direction across the shock
• Example wedge at angle of attack ? greater
  than wedge half angle ?
• This type of flow is called an expanding flow, in
  contrast to the oblique shock which creates a
  compressing flow.
• Instead of a shock, a expansion fan appears,
  which is comprised of infinite number of Mach
  waves called Prandtl-Meyer expansion waves
• Each individual expansion wave is isentropic
  flow across entire expansion fan is isentropic
• Ma2 gt Ma1
• P, ?, T decrease across the fan

Flow turns gradually as each successful Mach
wave turnsthe flow ay an infinitesimal amount
48
Shock Waves and Expansion WavesPrandtl-Meyer
Expansion Waves

• Prandtl-Meyer expansion fans also occur in
  axisymmetric flows, as in the corners and
  trailing edges of the cone cylinder.

49
Shock Waves and Expansion WavesPrandtl-Meyer
Expansion Waves

• Interaction between shock waves and expansions
  waves in over expanded supersonic jet

50
Duct Flow with Heat Transfer and Negligible
Friction

• Many compressible flow problems encountered in
  practice involve chemical reactions such as
  combustion, nuclear reactions, evaporation, and
  condensation as well as heat gain or heat loss
  through the duct wall
• Such problems are difficult to analyze
• Essential features of such complex flows can be
  captured by a simple analysis method where
  generation/absorption is modeled as heat transfer
  through the wall at the same rate
• Still too complicated for introductory treatment
  since flow may involve friction, geometry
  changes, 3D effects
• We will focus on 1D flow in a duct of constant
  cross-sectional area with negligible frictional
  effects

51
Duct Flow with Heat Transfer and Negligible
Friction

• Consider 1D flow of an ideal gas with constant cp
  through a duct with constant A with heat transfer
  but negligible friction (known as Rayleigh flow)
• Continuity equation
• X-Momentum equation

52
Duct Flow with Heat Transfer and Negligible
Friction

• Energy equation
• CV involves no shear, shaft, or other forms of
  work, and potential energy change is negligible.
  
• For and ideal gas with constant cp, ?h cp?T
• Entropy change
• In absence of irreversibilities such as friction,
  entropy changes by heat transfer only

53
Duct Flow with Heat Transfer and Negligible
Friction

• Infinite number of downstream states 2 for a
  given upstream state 1
• Practical approach is to assume various values
  for T2, and calculate all other properties as
  well as q.
• Plot results on T-s diagram
• Called a Rayleigh line
• This line is the locus of all physically
  attainable downstream states
• S increases with heat gain to point a which is
  the point of maximum entropy (Ma 1)

54
Adiabatic Duct Flow with Friction

• Friction must be included for flow through long
  ducts, especially if the cross-sectional area is
  small.
• Here, we study compressible flow with significant
  wall friction, but negligible heat transfer in
  ducts of constant cross section.

55
Adiabatic Duct Flow with Friction

• Consider 1D adiabatic flow of an ideal gas with
  constant cp through a duct with constant A with
  significant frictional effects (known as Fanno
  flow)
• Continuity equation
• X-Momentum equation

56
Adiabatic Duct Flow with Friction
57
Duct Flow with Heat Transfer and Negligible
Friction

• Energy equation
• CV involves no heat or work, and potential energy
  change is negligible.
• For and ideal gas with constant cp, ?h cp?T
• Entropy change
• In absence of irreversibilities such as friction,
  entropy changes by heat transfer only

58
Duct Flow with Heat Transfer and Negligible
Friction

• Infinite number of downstream states 2 for a
  given upstream state 1
• Practical approach is to assume various values
  for T2, and calculate all other properties as
  well as friction force.
• Plot results on T-s diagram
• Called a Fanno line
• This line is the locus of all physically
  attainable downstream states
• s increases with friction to point of maximum
  entropy (Ma 1).
• Two branches, one for Ma lt 1, one for Ma gt1

59
Duct Flow with Heat Transfer and Negligible
Friction